On an extension of a theorem on conjugacy class sizes.
type: Item
instance of: scholarly article
title: On an extension of a theorem on conjugacy class sizes.
zbMATH Open document ID: 1214.20030
author: Qingjun Kong
author: Xiuyun Guo
published in: Israel Journal of Mathematics
publication date: 2010-12-06T00:00:00Z
review text: The authors generalise a result of the reviewer in 1972 [J. Lond. Math. Soc., II. Ser. 5, 127-132 (1972; Zbl 0242.20025)] in which it was proved that if the conjugacy classes of a finite group have sizes precisely \(\{1,p^a,q^b,p^aq^b\}\) where \(p^a\) and \(q^b\) are prime powers then the group is nilpotent. Later \textit{A. Beltrán} and \textit{M. J. Felipe} generalised this in a series of papers [J. Group Theory 9, No. 6, 787-797 (2006; Zbl 1108.20021), J. Algebra 296, No. 1, 253-266 (2006; Zbl 1091.20017), ibid. 320, No. 12, 4317-4319 (2008; Zbl 1155.20306)] to prove that if the conjugacy classes of a finite group have sizes precisely \(\{1,m,n,mn\}\) where \(m\) and \(n\) are coprime then \(G\) is nilpotent. The authors consider what happens when there are only conditions are biprimary elements in a finite group. An element \(g\) of a group is called `biprimary' if its order is divisible by at most two primes. So the main result in this paper is: If the conjugacy class sizes of biprimary elements of a finite group have sizes precisely \(\{1,p^a,m,p^am\}\) where \(p^a\) is a prime power and \(m\) is prime to \(p\) and further there is a \(p\)-element whose conjugacy class size is \(p^a\) then \(G\) is nilpotent.
reviewed by: A. R. Camina
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 5823091
zbMATH Keywords: finite groups
zbMATH Keywords: conjugacy class sizes
zbMATH Keywords: conjugacy classes
zbMATH Keywords: nilpotent groups
zbMATH Keywords: biprimary elements
MaRDI profile type: Publication
full work available at URL: s11856-010-0082-1
OpenAlex ID: W2086873451
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DOI: 10.1007/S11856-010-0082-1
description: scientific article; zbMATH DE number 5823091